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# Hypersonic Aerothermodynamics AAE 51900

Purdue

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This 14 page Class Notes was uploaded by Robb McCullough on Saturday September 19, 2015. The Class Notes belongs to AAE 51900 at Purdue University taught by Steven Schneider in Fall. Since its upload, it has received 97 views. For similar materials see /class/207859/aae-51900-purdue-university in Aerospace Engineering at Purdue University.

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Date Created: 09/19/15

46th AIAA Aerospace Sciences Meeting and Exhibit AIAA 20081153 7 10 January 2008 Reno Nevada SurfacePressure Measurements of SecondMode Instability in Quiet Hypersonic Flow Malte Estorf and Rolf RadespielJf Technical University Braunschweig 38106 Braunschweig Germany Steven P Schneideri Purdue University West Lafayette IN 470971282 USA Heath B Johnson University of Minnesota Minneapolis MN 55455 USA Stefan Heinll DLR Institute of Aerodynamics and Flow Technology 37073 Go39ttingen Germany Surface pressure sensors have been used to measure the second mode boundary layer instability on a 7 half angle sharp cone at zero angle of attack in the quiet Mach 6 wind tunnel at Purdue and in t e conventional Mach 6 wind tunnel in Braunschweig The measurements were made using a stream wise array of high frequency sensors They show the second mode waves in quiet and noisy ow at different unit Reynolds numbers The quiet ow data is compared to results in noisy ow The signal quality allows for the calculation of ampli cation rates which are compared to the results of linear stability computations Nomenclature Latin symbols U ow velocity in test section a sonic speed I axial position A ow area f frequency Greek symbols l tube length iai logt growth rate per arc length M Mach number 7 ratio of speci c heats P tube pressure p uid density pressure uctuation amplitude R mass speci c gas constant Indices Re Reynolds number 0 initial quantity 8 arc length along surface 00 freestream quantity t time e edge quantity T temperature t tube stagnation quantity u ow velocity in tube sonic condition Research Assistant Member AIAA lProfessor Senior Member AIAA lProfessor Associate Fellow AIAA Senior Research Associate Senior Member AIAA lResearch Scientist 10f 14 Copyright 2008 by Steven P Schneider Published by the gi EWitif ermission I Introduction Much of the uncertainty in the aerothermodynamic design of hypersonic ight vehicles is due to the poor understanding and uncertain prediction of laminar to turbulent boundary layer transition The transition process is associated with the growth and nal breakdown of disturbances in the boundary layer ow Sev eral instability modes are known The disturbances they generate have been detected by a multitude of experimental techniques and predicted in numerical investigations Schlichting and Gersten1 and Schnei der2 provide comprehensive lists of literature on the subject However the mechanism based prediction of transition in hypersonic ow is an outstanding and demanding task2 In particular it is dif cult to obtain experimental data at well controlled conditions Among the multitude of unspeci ed in uences on hyper sonic transition the most well known is the acoustic noise that is radiated from the turbulent boundary layer on the nozzle wall in conventional hypersonic wind tunnels3 4 Therefore the comparison of instability measurements in noisy and quiet hypersonic ow is of particular interest In symmetric ow at hypersonic speeds the dominant instability is the second mode5 Some measurements of second mode waves have been carried out in the past with hot wires in both noisy and quiet ow2 However these measurements are very dif cult due to the limited mechanical strength of small hot wires with suf ciently high frequency re sponse Moreover the downstream in uence of the hot wire s support precludes simultaneous amplitude measurements at several stream wise positions Hence an appropriate non intrusive measurement technique is desirable Recently Fujii6 has shown that fast surface pressure sensors can measure the second mode on a cone in noisy hypersonic ow In the present work this technique is used to measure the second mode under quiet ow conditions in the Boeing AFOSR Mach 6 quiet tunnel of Purdue University Those measurements are compared to measurements under noisy conditions in the Purdue tunnel and in the Mach 6 hypersonic facility at Technical University Braunschweig II Experimental Setup IIA The BoeingAFOSR Mach6 Quiet Tunnel All Clean Stainless Steel From SecondThroat Section Upstream Unique LowNoise Flow olue to Laminar NozzleWall Boundary Layer Slow Gate Valve BleeoISlot Suction Plumioeol Botn T rough Fast Valve to Tank anol to DiFFu er 487 EEE gzzm Contraction Ainoiowsf quot quot quot 39 Max 300 psig 817 Ioaraos anol 398F Fixed Sting Support 800C IIIne 68s DiFFUSer r n er no r AEOUEC 10Puun Double Burst Diaphragm 175in Driver KTuloe 1885 Ft long 4000 CuMc Ft Vacuum Tank ODE V OTMQ COST Sliding Sleeve Figure 1 Schematic of the BoeingAFOSR Mach6 Quiet Tunnel The BoeingAFOSR Mach 6 quiet tunnel at Purdue University is a blowdown facility designed as a Ludwieg tube as sketched in Fig 1 The 374 m long heated driver tube on the left serves as a pressurized air reservoir which discharges through the converging diverging nozzle into the vacuum tank Before each run the pressurized assembly is separated from the vacuum tank by a pair of diaphragms placed downstream of the test section When the diaphragms burst an expansion wave travels upstream through the test section into the driver tube There it re ects forth and back and changes the state of the air each time it passes This causes the ow condition in the test section to change approximately every 200 ms when the wave reaches the nozzle Fig 2 shows the stepwise change of pressure as measured at the contraction wall7 Fig 2 also shows measurements from a Pitot probe in the test section The Pitot pressure shows the sudden change from noisy to quiet ow condition when the contraction pressure falls below a certain pressure limit here 145 psia The quiet ow is achieved only by extensive care in design and construction including a large driver tube and a highly polished nozzle with bleed slots for the contraction wall boundary layer8 Although 20f14 American Institute of Aeronautics and Astronautics 3 3 3 33 3 3 3 3 PItot pressure pSIa 3 Contraction pressure psia 150 3 71 8 6 140 g 130 g 3 l 3 a 9 i I g o I E 4 3 3 3 3 3 3 33 45 4 11 39 l 110 9 CL I 3 E E imisyiinterniitienttIIrb f 3 39i100 8 39 39 39 39 39 3 W as0tSonnozwa 4 3 339 39 290 2 I I I l I I I l I I I I II I I I I I I I I I I l I I I 1 0 1 2 3 4 5 6 7 TImes Figure 2 Sample of contraction pressure and Pitot pressure in BoeingAFOSR Mach6 Quiet Tunnel the quiet performance degraded in late spring 20079 it was restored in the late summer by repolishing the throat10 When the measurements presented in this paper were made the tunnel was running quiet up to stagnation pressures of 145 psia By closing the bleeds for boundary layer suction at the nozzle throat the tunnel can be operated in noisy mode with turbulent nozzle wall boundary layers The tunnel can be operated at any pressure up to 270 psia IIB The Hypersonic Ludwieg Tube Braunschweig A schematic of the hypersonic Ludwieg tube Braunschweig HLB is given in Fig 3 In this facility the driver tube is separated from the low pressure section by a fast acting valve Only the first 3 m long section of the driver tube is heated which suits the amount of gas that escapes during one run The valve consists of a streamlined body on the tube axis with a pneumatically driven piston that plugs into the nozzle throat The valve opens within 20 ms and closes after the expansion wave has passed back and forth within the tube that is about 80 ms after full opening A detailed numerical analysis has been made for the onset of flow and the establishment of steady flow conditions in the test section11 The nozzle maintains an opening half angle of 30 which results in slightly expanding flow in the test section with Mach numbers between 58 and 595 depending on the axial position and on the unit Reynolds number The latter can be chosen by the initial driver tube pressure between 3 X 106771 and 20 X 106777 More details on the HLB were reported earlier 12 13 Fast Acting Valve Diffuser Storage Tube Test Section Dump Tank Heating Laval Nozzle Figure 3 Schematic of the hypersonic Ludwieg tube Braunschweig IIC Model and Instrumentation Measurements in both facilities were made on the same 70 sharp cone of 495 mm length The model was made from Plexiglas to allow future infrared heat transfer measurements on the same cone The apex was made from steel in order to provide better mechanical strength The accuracy of the tip was visually checked under a microscope The nose radius was measured with a stage micrometer as about 001 i 0003 mm The model was instrumented with a stream wise array of high frequency pressure sensors of type PCB M131A32 The 30f14 American Institute of Aeronautics and Astronautics sensors were ush mounted in the model surface at three axial positions I 340 mm 365 mm 0390 mm for the experiments in Purdue and at four positions I 315 mm 340 mm 365 mm 390 mm for the experiments in Braunschweigi The diameter of the sensing area of those sensors is 318 mmi Power was supplied to the pressure sensors using an instrument supplied by the manufacturer PCB 482A22 which at the same time also performed signal conditioning According to the manufacturer s speci cation the resonance frequency of the pressure sensors is larger than IMHz and the output signal is highpass ltered at 10 kHz The sensors are calibrated in a shock tube by the manufacturer and have sensitivities between 150 and 167mVpsii For details on the the measurement technology the reader is referred to the manufacturers websitei1 IID Data Acquisition The data acquisition in Purdue was performed with an oscilloscope Tektronix DPO 7054 at a sampling rate of 4 MSsi The voltage range was chosen as i5 mV for runs with quiet conditions and i2 mV for noisy runs The oscilloscope was operated in a 77HighRes77 mode where it samples the data internally at the maximum sampling rate and averages the data on the y into memory According to the manufacturer this mode gives an effective resolution of 11 bits at 4 MSs compared to 8 bits without averaging and at the same time lters high frequency noise In this mode the oscilloscope writes the data in a 16bit format 2 bytes According to the sensor calibration this gives an output resolution of 10 6 psi using the quiet mode settings The scope has 50 megabytes per channel of memory In Braunschweig a 16 bit transient recorder PCI express card Spectrum M2ii4652 was used for data acquisition with a PC The card allowed a maximum sampling rate of SMSsi The ampli cation of the card was set to measure a voltage range of iSOmVi IIE Model Installation In the Purdue quiet tunnel the base of the cone was installed 71 mm upstream from the end of the nozzle on the axis of the tunnel Due to the tight fabrication tolerances of the sting support section and the snug sliding t of the sting itself the cone was assumed to be well aligned with the nozzle axisi Hence the angle of attack was not checked before the experiments After the test series a single measurement was taken with the cone model turned by 180 around its base support plate The ow condition in that run was noisy and evaluation showed later that the boundary layer was already transitional at the sensors However the maximum 2nd mode frequency in that run was about 10 kHz smaller than in a run at identical conditions with the model turned by 180 not shown in this paper This is likely due to a small angle of attack with the sensor array being slightly windward in all measurements presented in this paper at quiet conditions For the Braunschweig measurements the cone was installed with its apex 50 mm upstream from the beginning of the test section The cone axis was positioned 50 mm above the axis of the tunnel in order to avoid a conical compression wave that is known to focus on the axis within the test section15 The cone was aligned to the ow by measuring the line of transition with an infrared camera The alignment was improved in subsequent tunnel runs until the transition line was found to be straight on the cone to within ilO mm from two perpendicular view anglesi III Determination of ow conditions IIIA Purdue experiments The drivertube pressure was measured at the entrance of the contraction during all tunnel runsi A temper ature measurement was taken with a thermocouple at the end of the driver tube before every run However the temperature within the tube is strati ed and there are also local differences in the tube temperature due to nonuniform heating and insulation16 To improve the estimate of the mean temperature in the tube the trace of the pressure was used to nd the temperaturedependent velocity of sound that corresponds to the frequency of the expansionwave re ections within the tube For this the ow of the air out of the tube was modeled by two characteristic lines corresponding to the head and the tail of the expansion wave as shown in Fig 4 The details of the calculation are given in the appendix A formula for the temperature at X 0 after n re ections of the wave is given there as Eq The traveling times of the head and tail of the wave are given in Eqs 10 and 11 respectively as functions of the tube Mach number M Applying the isentropic relation PPo TToYVI1 to Eq 7 shows that the pressure ratio between 4of14 American Institute of Aeronautics and Astronautics u20 Mt gt 150 v 430 I l l A t l l l 140 l x 420 IE 1 w I t L i a l g 130 l 410 393 c a l 5 I l B2 2 a L g S L O H A2 39 5120 400qEJ g l 01 B1 l H l Pcalculated T calculated 0 100 39 39 39 39 39 39 380 gt 0 1 2 3 4 5 CC time t s a characteristic diagram b sample evaluation Figure 4 Illustration of driver tube gas condition modeling for Purdue quiet tunnel subsequent wave re ections is only a function of the tube Mach number Hence with the measured pressure trace the tube Mach number can be estimated to t the height of the pressure steps Finally the length of the steps can be tted to the measured trace by adjusting the estimate of the initial mean temperature in the tube These two estimates have been performed for all tunnel runs Figure 4b shows a pressure trace recorded during a quiet run with initial pressure of 1498psia and the corresponding calculations for the pressure and the temperature The tube Mach number was Mt 00046 during the rst 27s and changed to Mt 00044 after that period The initial temperature of the gas assumed for the calculations was T0 434K which agrees with the temperature measured before the run However if the temperature for the calculation is changed by 1 K there is already an offset of about 0007s between the calculated and the measured trace after 6 seconds of the tunnel run which is about twice the accuracy of the t This indicates that the procedure described above is a reasonable check for the measured temperature In fact the temperatures found by this procedure varied between T 0 434K and T0 440K while the thermocouple did not show this variation The calculation of pressure and temperature allows for an accurate Reynolds number determination throughout the whole run A linear viscosity law was assumed17 with the viscosity taken as u 766i0005 X 10 6 at 1104 K The uncertainty of the air temperature is estimated to i08 and the uncertainty of the determined pressure is about 02 The Mach number in the test section is known from previous Pitot measurements to be M 58 in noisy operation and M 60 in quiet ow with an estimated uncertainty of i2 This enables calculating the Reynolds number in the test section with an uncertainty of about i4 When operated quietly the nozzle wall boundary layer remains laminar producing free stream noise levelsa on the order of 005 When operated conventionally the nozzle wall boundary layer is turbulent and the noise level increases to about 3 IIIB Braunschweig experiments In Braunschweig the driver tube pressure is measured before each run with an accuracy of 1 The ratio of initial pressure P0 to the total pressure after the expansion P175 is well known from a large number of previous measurements POP17 0932700007 The driver tube temperature is measured during each run by two fast thermocouples These penetrate into the ow on the upper and lower side of the valve section and enable calculating the mean temperature of the gas during a run However the temperature difference between the upper and lower measurement position is as high as 30K and the mean of those temperatures is not proven to be the total temperature at the height of the model in the test section This results in an estimated uncertainty of ll in the total temperature The Mach number in the test section has been Ppit ppit2 ppit aNoise levels are de ned here as normalized rms of Pitot pressures 50f14 American Institute of Aeronautics and Astronautics determined by extensive Pitot measurements and by RANS calculations of the nozzle owiu 1n the range of unit Reynolds numbers between 478 X 106 it is M 585 i 0103 at the position of the pressure sensors This enables calculation of the Reynolds number in the test section with an uncertainty of about i2i The nozzlewall boundary layer in the tunnel is always turbulenti The noise level is between 1 and 115 depending on the unit Reynolds number based on preliminary measurementsi IV Data processing For each tunnel run in Purdue the oscilloscope recorded 45 seconds of pressure signal during the run and 05 seconds of reference data before the start of the tunnel From the calculated pressure and temperature the corresponding trace of the Reynolds number was calculated throughout the run For certain Reynolds numbers at different times during each run a period of 04s 1600000 samples was extracted This period was divided into overlapping windows with 2000 samples each The offset between the windows was 200 samples giving about 8000 windows These were multiplied with a normalized Blackman window and Fourier transformed The absolute values of the complexconjugate amplitudes were added and averaged over all 8000 windows In the same way an 0 35s period of the data measured directly before the start of the tunnel was processed giving only 7000 windows The power spectra of both transforms were subtracted Thereby the uncorrelated electronic background noise and the permanent spectrum of disturbances from the ambient 39 39 was J from the measured signalsi For some lownoise experimental data the power spectral subtraction of background noise yielded negative amplitude values at some frequencies These values were set to zero It is not clear whether this preliminary method of correcting the noise is the best or most appropriate one The data processing for the experiments in Braunschweig was essentially the same but due to the short run time of the tunnel only a period of 60 ms 180000 samples was evaluated from each run The window size for the FFTs was chosen as 1500 samples each The secondmode ampli cation rates iai were calculated from 1 A A iai if 1gt0 and p2gt0 1 iai 0 else 11 1 11 l with 151 and 132 being the pressure after noise at two sensors and 32 7 31 being the surface distance between those sensors 2512 mm V Stability Computations Stability computations using linear stability theory LST were made in order to compare the resulting local ampli cation rates to the measured resu tsi The computations were based on both full Navier Stokes solutions of the boundary layer ow and on similarity solutions for the compressible boundary layer The computation methods are described here brie yi VA Computations based on Navier Stokes solutions Laminar mean ows for the test cases were generated using an optimized ZDaxisymmetric mean ow solver based on the implicit DataParallel Line Relaxation DPLR method18 which is provided with the stability code STABLilg The solver produces secondorder accurate laminar ow solutions with low dissipation and shock capturingi For each of the simulations 360 X 360 point structured grids were generated with clustering both at the body surface and at the nose The ow for these cases was considered to be a nonreacting mixture of 767 N2 and 233 02 by mass Freestream conditions were obtained by applying the isentropic ow relations from the stagnation conditions to the speci ed freestream Mach numbersi Mackls viscosity model was used for the gas mixture5 The LST analysis of the cases was performed using the PSE Chem code which is distributed as a part of the STABL suiteilg For the LST analysis a parallel ow assumption was made by neglecting derivatives of mean ow quantities in the direction of the computational coordinate along the body Spatial ampli cation rates of disturbances were found for given disturbance frequencies and surface locationsi 60f14 American Institute of Aeronautics and Astronautics VB Computations based on similarity solutions Alternatively a simpli ed approach has been used to compute the basic flovv this neglects the transverse curvature of the cone and the interaction of the boundary layer with the shock wave20 Based on these assumptions the boundary layer on a sharp cone at zero angle of attack in supersonic flow can be obtained from a flat plate boundary layer at the same edge Mach number and temperature using the transformation of Mangler and Stepanov21 The flat plate boundary layer was calculated from self similar zero pressure gradient planar compressible boundary layer equations for a calorically perfect gas For the temperature dependence of the dynamic viscosity a modified version of the Sutherland lavv according to Mack5 was used The Prandtl number was assumed to be constant and set to P7 072 An isothermal vvall temperature of 315 K was chosen The boundary layer edge conditions were derived from the free stream conditions of the experiment using the shock relations and the inviscid solution for aXisymmetric supersonic conical flovv22 For the instability analysis of this self similar boundary layer flow the linear version of the NOLOTPSE code23 was used The NOLOTPSE code was developed in cooperation between DLR and F01 and solves compressible parabolized stability equations PSE formulated in curvilinear orthogonal coordinates for a thermally ideal gas Alternatively the code can also be used as a classical local stability analysis tool In the latter case an eighth order system of ordinary differential equations is solved assuming a locally parallel flow A more detailed description of the NOLOTPSE code was reported in 24 VC Computations T vvo reference cases were defined for the computations as listed in Table 1 The table also gives edge quantities that have been used to plot normalized results For the calculations the wall temperature was Table 1 Conditions chosen for stability analysis of 70 cone boundary layer and values for normalization M Ptlpsia Tth TwaulK Reesllm Helms casel 60 1106 418 315 1045 x 106 846 case2 58 893 415 315 917 X 106 839 chosen to be slightly more than ambient temperature Successive tunnel runs over the course of a day may have heated the model more but temperature measurements have not been taken A comparative stability calculation of case but with 325K model temperature resulted in only small deviations from the results shown here Only 2D waves have been considered for the stability calculations As a representative result Fig 5 shows the amplification rates calculated from the similarity solutions of case The normalized plot in Fig 5b shows that the frequency of the second mode scales well with UeVR ees This will be used for comparison of the experimental results at different unit Reynolds numbers in the following section 600 007 x 3906i S I 1m Ree s 0 CD 0 005 g 400 30 3 3EO6 f 004 gt gt o 300 o c C g 20 003 2E 06 E E quot 200 quot6 10 03902 1 1E06 quotC 9 100 001 O 0 I I I I O i I I I I 00 01 02 03 04 0 1E06 2E06 3E06 4E06 5E06 s m Ree a Stability diagram b Stability diagram non dimensional Figure 5 LST results based on similarity solution with appropriate nondimensionalization caseI 7of14 American Institute of Aeronautics and Astronautics VI Results VLA PressureFluctuation Spectra Figure 6 shows samples of pressure uctuation spectra at three consecutive sensors The data in Figure 6a was collected at Purdue under quiet ow at a 89r3psia drivertube pressure This was the lower limit of the pressure range where the second mode could be detected under quiet owr Note that the amplitude of the uctuations at that pressure is close to the limit of the available resolution 10 5 psi at 16 bit Therefore the signaltonoise ratio is very small and only the large number of averaged FFTs 8000 windows brings out the 2nd mode peaksr Also note that the resolution of the pressure sensors is speci ed as 10 3 psi in the manufacturer s datasheetr Moreover the diameter of the sensing surface 318 mm is larger than half A n21 Purdue E US 7 quietcunditmn BEDS quietcunditmn 39 n paaapsia pf EIBpsia n E RegtltB52EEm E E 340 mm s g A X 3B5 mm a BEBE e o X 39m mm g 495 S1 A 1 g A mung curves g 2 z g a A a g 4EVEIE e of A g ZEUS o 2 3 J A AA A A incll 39 0 I quotA 391 A 39 A N A AA A A A 339 o h o A A o o IA A A A 1 D 7 iuuuuu zuuuuu auuuuu ADDEIEIEI suuuuu u WEIEIEIEIEI zuuuuu auuuuu ADDEIEIEI suuuuu frequencytp tz frequencytp tz a measurement and curve t b measurement and calculation Figure 6 Spectra of pressure uctuations measured in quiet Mach6 ow quot R M E n RunSE Purdue un rauns WEig E39 DDS 39 HEHSy Eundmun E39 DDS on 82 3 psia l A39 T 4iB E mm A 3 RegtltB52EEm E mm a u 34m mm a A g A A A X 365 mm A a A t X 39D mm a A E Danae A o E nuns g uf 39 g E g E s 39 s r E H mm E H mm 3 A S o o H mm H um i i i i D iuuuuu zuuuuu auuuuu ADDEIEIEI suuuuu WEIEIEIEIEI zuuuuu auuuuu ADDEIEIEI suuuuu frequencyfHZ frequencyf Hz a noisy condition Purdue 13 noisy condition Braunschweig Figure 7 Spectra of pressure uctuations measured in noisy Mach58 ow the wavelength of the secondmode waves where half the wavelength can be approximated by the boundary layer thickness of 2 2 mmr For these reasons the absolute amplitudes given in the gure are questionable and have to be understood as some uncertain nonlinear function of the actual uctuation amplitude at a certain frequency However the measured amplitude is assumed to be linearly proportional to the mean amplitude of the uctuations when ampli cation rates are calculated in this paper The spectrum shown in Fig 6b was measured at 1106 psia drivertube pressure and shows a secondmode amplitude that is an order of magnitude higher The gure also shows integrated ampli cation rates from the linear stability calculation of casel based on the similarity solution The calculated amplitudes are scaled to match the amplitude at the second sensor position The measured ampli cation seems to be somewhat higher than the calculation The calculated frequencies are lower than those measured by about 8 kHz This frequency shift may be due to the suspected angle of attack discussed in section llrEr However the bandwidth of ampli ed frequencies is almost the same in the measurement and the calculation 80f 14 American Institute of Aeronautics and Astronautics Figure 7 shows spectra in noisy ow measured at the same Reynolds number as in Fig 6a at similar stagnation conditions In noisy ow the measured pressure amplitudes are about 450 times higher The secondmo e peaks in noisy ow are much broader and rst harmonics can be detected The small peak at about 310 kHz is thought to be an effect of the sensors although this is far below the resonant frequency claimed by the manufacturer At this Reynolds number in noisy ow the boundary layer appears to be transitional since the second mode amplitude growth stagnates and reverses between the second and third streamwise sensorl Note that the results in the Braunschweig tunnel show the same behavior at nearly the same Reynolds number and wave amplitude Fig 7b Since no data was collected at Purdue at lower unit Reynolds numbers than shown in Fig 7 we can compare data at the noisy condition only as described with the Braunschweig experiments in the following section 0 Re 2 21E6 Run35 noisy o Rex 2 54E6 Run35 noisy III Rex 2 22E6 Run 21 quiet fluctuation magnitude p psi D i i 0 500000 iE06 i5E00 nequency t HZ Figure 8 spectra of pressure uctuations measured in noisy and quiet ow at same Reynolds numbers Figure 8 plots the data of Figs 6a and 7a together using a logarithmic scale The logarithmic plot shows that in the transitional boundary layer both the lower and higher frequency portions of the spectrum are starting to ll in However it is not yet possible to determine how much of the difference in spectra is due to nonlinear ampli cation and breakdown of the waves and how much is due to the difference in the spectrum of the freestream noisel Since under quiet ow at this pressure the second mode is just becoming detectable the measurement technique does not permit comparing linear ampli cation rates at the same Reynolds number in quiet and noisy hypersonic owl It will be necessary to use controlled perturbations in the quiet ow boundary layer to make such a comparison 1 O ReX 225E6 Run11 Braunschweig o ReX 095E6 Run09 Braunschweig 151 t A Rewx 48E6m Pitot measurement 0 0 normalized pressure amplitudes F I 200000 4 frequency f Hz I 00000 600000 Figure 9 Surface pressure uctuations normalized by calculated edge pressures at two di erent Reynolds numbers compared to normalized Pitot pressure uctuations measured in the Braunscbweig tunnel 90f 14 American Institute of Aeronautics and Astronautics Figure 9 shows normalized pressure spectra measured at the cone surface at two different Reynolds numbers compared to a measured Pitot pressure spectruml The surface pressure spectrum at the higher Reynolds number is the one from Figure 7b at z 0mm which is just before stagnation of the second mode growthl Note that the Pitot spectrum was measured with an absolute pressure sensor providing the actual mean pressure for normalizationl Whereas the surface pressure spectra were normalized with the edge pressure calculated from inviscid conical ow solution at M 58 for the measured stagnation pressurel Therefore the normalization of the surface pressure spectra is somewhat uncertainl However the normalized surface pressure uctuations are by almost a factor 2 higher than the Pitot uctuations already at low frequencies The reason for that difference is not clear An effect of receptivity or an early ampli cation of disturbances within the cone boundarylayer can be suspected en rms values of pressure measurements are used to compare noise levels although this quantity does not give any information on the spectral distribution of the uctuations However in order to give some idea of how much the rms of the uctuations rises before transition occurs it may be of interest to provide the rms values for the three measurements shown in Figure 9 rmsA 14 rmsO 67 rmsltgt 13l VLB Ampli cation rates it Ree n V g 3 056 Slmll E S 7 4 8E6 Slmll 3 06 gt 3 056 Nav St dquot 4 8E6 Nav St 6 if 315E6 E 3E706 g 2506 E 8 8 2506 E 1506 E T 1506 E 0 E E E 0 E E m 4506 m 7506 or a c 2 2 I i l 0 2 003 004 005 006 0 7 0 0 00 005 reducedfrequencyf sUEReg39Z reducedfrequencyf sUEReg a quiet condition Purdue 13 noisy condition Braunschweig Figure 10 Ampli cation rates as calculated from measured pressure amplitudes compared to LST values based on Navier Stokes solution and on similarity solution Fig 10 shows the logarithmic ratio of the amplitudes normalized by the unit edge Reynolds number vs the reduced frequency in both quiet ow a and noisy ow The results shown for the quiet cases are averages over two to four datasets from different runs at the same Reynolds number whereas the results shown for noisy data are from single tunnel runs The scattering of the data in quiet ow at the low frequency edge of the ampli ed spectrum is caused by taking a ratio of weak signals with very low signalto noise ratio Nevertheless the trend of that data shows that the measured quiet ow bandwidth for the ampli ed waves is narrower than predicted by the calculations Moreover the ampli cation rates of the quiet cases are smaller than predicted by linear stability theory However both shortcomings may be due to the suspected angle of attack see section KB or possibly to the relatively large sensor size The measured maximum ampli cation rates in noisy ow agree well with calculations at the lower Reynolds numbers However even at the smallest Reynolds numbers measured the measurements show ampli cation at lower frequencies than the computations For edge Reynolds numbers higher than 22 X 106 the maximum ampli cation rates rapidly decreasel This is in accordance with Stetsonls observations25 who reported this behaviour at Mach 8 for iRee gt 1400 Stetson also reported a growth of harmonics of the second mode at higher Reynolds numbers which he suspected was due to nonlinear effects But looking at the ampli cation rates of higher frequencies in Fig ll shows that for all Reynolds numbers measured in noisy ow the ampli cation rate of the harmonics is the same as the fundamental ampli cationl Hence it seems that this is not necessarily an indicator for nonlinear growth but just a result of the pressure signature on a sensor s surface being more complex than a pure sine wave Nevertheless comparing the ampli cation rates 10 of 14 American Institute of Aeronautics and Astronautics 5506 E 2 4506 we 95 3506 8 2506 E 3 1506 E E 0 E E N 71E706 c 9 H l l l l 0 l l 001 002 003 004 005 009 01 011 012 013 04 7 0 08 reducedfrequencyf sUERe g Figure 11 Ampli cation rates as calculated from measured pressure amplitudes extended spectrum from Fig 10b measured in noisy ow to those measured in quiet ow there is a signi cant difference in the bandwidth of ampli ed frequencies which may be due to nonlinear effects even at the lowest measured Reynolds numbers For the given cone the lowest feasible Reynolds number was limited by the operational range of the facility in Braunschweigl At those conditions in noisy ow the second mode signal was still 50 times larger than the weakest secondmode wave that was detectable under quiet owl ln Stetson7s26 experiments with hot wires at Mach 8 the secondmode was rst detectable at Ree m 12 X 105 which is close to the smallest Reynolds number analyzed here in noisy owl VII Conclusion The use of high frequency surfacepressure sensors for measuring pressure uctuations caused by second mode boundary layer instabilities has been proven to be very effective even in quiet hypersonic owl T e advantages of this technique over hotwire measurements for determining streamwise ampli cation rates are L No downstream in uence of the sensors is apparent and therefore streamwise arrays of sensors can be used to easily access spatial ampli cation rates for instabilitiesl This is an easyto use offthe shelf measurement technology at comparatively low cost The sensors used are very resistant to the harsh environment typical of hypersonic facilitiesl F90 The high frequency response of the sensors allows for measurements up to the 1 MHZ ranger l The sensitivity for detecting the second mode in terms of signalto noise ratio may be an order of magnitude better than in the hotwire experiments by Stetson et all although Stetson et all used completely different data acquisition systems 01 These very promising preliminary results suggest that the limited spatial resolution of the surfacepressure sensors might be alleviated through care 1 comparison to spatial averages of the computational resu tsl The secondmode amplitudes under noisy conditions were 450 times higher than under quiet ow at the same Reynolds number This ratio of the secondmode amplitude seems to be much larger than the ratio of the broadband Pitot uctuations which was about 50100 perhaps due to the spectral content of the freestream noise and also to receptivity effectsl Under noisy conditions the secondmode amplitudes began to decrease at about the same Reynolds number in both the Purdue and Braunschweig tunnels which suggests that results in the two tunnels can be compared to enable the study of nonlinear effects Under quiet ow at this same Reynolds number the instability amplitudes just began to rise from the measurement noise Therefore it was not possible to make a direct comparison of second mode ampli cation rates in noisy and quiet owl Measured maximum ampli cation rates were in good agreement with those calculated using linear stabil ity codes In addition the bandwidth of ampli ed frequencies was in good agreement under quiet owl T e 11 of 14 American Institute of Aeronautics and Astronautics remaining differences are thought to be due to a slight angle of attack in the quiet experiments However under noisy conditions the computed bandwith of ampli ed frequencies was more broad than the measure ments for all Reynolds numbers down to Ree 15 X 105 The measured noisy ow maximum ampli cation rates were in good agreement with calculations below Ree 2 2 X 105 as in Stetson s observations at Mach 8 uture measurements should check the angle of attack more carefully by comparing the frequencies of the mostunstable wave on opposite sides of the cone It would also be very interesting to compare measurements of the instability waves using surfacepressure sensors to measurements using a hot wire or laser differential interferometeri A series of measurements with the surface pressure sensors under different conditions might be combined with computations and measurements using other instrumentation in order to determine the effect of the size of the pressure sensor The pressure sensors can then be used in several tunnels to compare 2ndmode wave growth under conditions where hot wires will not survive A Appendix Figure 4a shows an idealized characteristic diagram of the ow in the tube shortly after the start of the tunnel The expansion wave starting at the nozzle at t 0 is modeled by two characteristic lines corresponding to the head and the tail of the wavebi The wave is re ected at the boundaries after each pass through the tube The boundary conditions at the ends of the tube are found as follows At the throat an unknown and time dependent critical mass ux per area a p can be assumed with a being the velocity of sound at critical condition and p being the critical densityi When A is the combined throat area for the main ow and the bleed slot the corresponding mass ux per area at the tube diameter A is pu p a A with p and u being the areaaveraged density and velocity at diameter A The ratio of the mass uxes per area is related to the ow Mach number ME by the equation for a onedimensional stationary isentropic expansion 1 pu 7 lt2lt771ME2gt77 157A 1 t Y 1 A Therefore the Mach number in the tube is a function only of the area ratio and independent of the state of the uid in the tube Hence the boundary conditions of the characteristics for t gt 0 become u Mt a at the right end of the tube which is at z 0 assuming the contraction length is small compared to the tube length and u 0 at the upstream end where z fl with 1 being the tube lengthi With these assumptions the state of the gas in areas I to II and after all subsequent passes of the expansion wave can be calculated as a function of the tube Mach number by the following procedure However the exact tube Mach number is not known in advance due to the unknown effect of the boundary layers displacement thickness on the effective tube diameteri Therefore the Mach number has to be determined from the measured pressure drop after each re ection of the expansion wave The head of the expansion wave travels with the velocity of sound a0 into the gas So the slope of the characteristic line in the diagram is fi f 7040 before its re ection at A1 Fig 4a At point B1 the head of the re ected expansion wave intersects the tail of the incident expansion waver Between A1 and B1 its slope changes to a 1 Mt according to the state of the gas in area I ful lling the downstream boundary condition The conditions in area I can be calculated by integrating the compatibility equation for the C characteristic from A1 to 1 aIM I 7 du V 7 0 p a 7 1 0 2 WM fie 7 ll A lt3 1 a 7 T 7 27 and rearrang1ng w1th a 1 7 g1ves which gives o T 7 7 71 72 To 7 1 TMEgt i 4 bUsing more than two characteristic lines would complicate the calculation However the very low Mach number in the driver tube results in weak waves that are well captured with only two characteristic lines 12 of 14 American Institute of Aeronautics and Astronautics The pressure and density ratios can be calculated from 4 using isentropic relationsi22gt27 The initial slope of the tail of the expansion wave is a M 7 1 before its re ection in C1 and a1 thereafter The state of the gas in area H is calculated by integrating the compatibility equation for the C characteristic from B1 to C1 0 T II du 7 1th 7 i 1 I xT Performing the integration and using Eqs 3 and 4 gives after rearrangement 7 31 8135 To 7 The state in area I can be found by integrating from B2 to C2 Since the uid in area H is at rest its condition can be regarded as the new initial state TH To Then the integration from B2 to C2 is similar to that from A1 to A2 and the relations for all subsequent re ections are given by Eqsi4 and So nally the static temperature at the contraction after time tag with n 12 3 i i changes to 5 6 Hwy 7 2 7 7 1Mt The time needed for the head of the wave to travel forth and back in the tube can be calculated from the slope of the characteristic lines I l I 2 g lM tM jr M a0 a11 2 a0 1 z 1 2 T2n1 To 1 WTIM The time between the arrival of the head and tail at z 0 is 7 l 2771Mt71 24wmm Adding the traveling times of the head for subsequent re ections using Eq 6 gives 1 wig 1M 1 2771Mt k mn 3lt 1Mz 1l27w71Wzl 10 Accordingly the time lag between the head and the tail of the wave after several re ections becomes 1 2771ME 2771ME 1 2771M to 2quot ao l27 v71Mz M 17 M 1l 39 l1 T l2 7 v71le 11 Note that for the calculation of these times due to the modeling by only two characteristic lines a step change of the traveling speed was assumed at the boundaries rather than a continuous change within the overlapping areas ABCI So the above formulas will underestimate the travel time for the head and overestimate that for the tail of the wave Therefore the exact path of the expansion wave will be somewhere between the calculated characteristics Hence the uncertainty of the path is given by somewhat less than half of the distance between head and tail For the given tube with a length of 1255 feetC and a Mach number Mt 00046 at temperature To 433K the time lag between head and tail can be calculated for n15 from Eq 10 It is to30 7 tAgo 00153 which is about 05 of the overall traveling time at that point Acknowledgments The rst author would like to thank Professor Schneider s research group for their great support during his 3 month stay at Purdue Dri Ki Fujii of JAXA was kind enough to give full information regarding the sensors used in his experiments and also by the present authors Operating costs for the Purdue tunnel were funded by AFOSR under grant FA95500610182i cThe length of the tube is 1225 feet and the contraction has a length of 40 inches The last 4 inches of the contraction have been neglected in the calculation since the sudden rise of the Mach number will cause the upstream information to fade somewhere close to the throat 13 of 14 American Institute of Aeronautics and Astronautics References 1Schlichting H and Gersten K Boundary Layer Theory Springer Berlin 8th ed 2000 2Schneider S P Hypersonic laminareturbulent transition on circular cones and scramjet forebodies Progress in Aerospace Sciences Vol 40 2004 pp 10 Pate S R and Schueler C J Radiated Aerodynamic Noise Effects on BoundaryeLayer Transition in Supersonic and Hypersonic Wind Tunnels AIAA Jounal Vol 7 No 3 1969 pp 4507457 4BeckWith I E and Miller III C G Aerothermodynamics and transition in highespeed Wind tunnels at NASA Langley Annu Rev Fluid Mech Vol 22 1990 pp 419439 5 L M BoundaryiLayer Linear Stability Theory Special Course on Stability and Transition of Laminar Flow AGARD Special Course at the yon Karman Institute RhodeiSaintiGenese Belgium March 26730 1984 AGARD Report No 709 1984 pp 37173781 6 jii Experiment of TwoeDimensional Roughness Effect on Hypersonic BoundaryeLayer Transition J Spacecraft and Rockets Vol 43 2006 pp 7317738 7Juliano T J Swanson O and Schneider S P Transition Research and Improved Performance in the Boe ingAFOSR Mach76 Quiet Tunnel AIAA Paper 200770535 2007 8Schneider S P The Development of Hypersonic Quiet Tunnels AIAA Paper 200774486 2007 9Schneider S P and Juliano T J LaminareTurbulent Transition Measurement in the BoeingAFOSR Mach76 Quiet Tunnel AIAA Paper 200774489 2007 B r Schneider S P and Juliano T J Effect of F reestream Noise on Roughnesselnduced Transition for the X751A Forebody AIAA Paper 200amp0592 2008 s and Radespiel R Investigation of the starting process in a LudWieg tube Theoretical and computational Fluid Dynamics Vol 21 2007 pp 81798 Estorf M Wolf T and Radespiel R Experimental and numerical investigations on the operation of the Hypersonic LudWieg Tube Braunschweig 5th European Symposium on Aerothermodynamics for Space Vehicles 2004 13Estorf M Radespiel R Heine M and quot11 R Der quot i quot I 391 HLB DGLReJahrbuch 2003 2003 pp 6617670 14PCB Piezotronics Pressure and Force Sensors Division Pressure Catalog httpWWWpcbcomLinkedDocuments PressurePFScatpdf 2007 15E torf M Ortsaufgelo39ste Messung 39 39 quot VI quot 39 in der A i i i y il Eingereichte Dissere tation Fakultat fur Maschinenbau TU BraunschWeig 200 16Borg M P Schneider S P and Juliano T J Inlet Measurements and Quiet Flow Improvements in the Boe ingAFOSR Mach76 Quiet Tunnel AIAA Paper 200671317 2006 17Fiore A W Viscosity of Air J Spacecmft and Rockets Vol 3 1966 pp 7567758 18V7right M J Candler G V and Bose D A DataeParallel LineeRelaxation Method for the NaviereStokes Equations Paper 9772046CP AIAA 1997 19Johnson H B and Candler G V Hypersonic Boundary Layer Stability Analysis Using PSE Chem Paper 200575023 AIAA June 2005 20Simen M and Dallmann U On the instability of hypersonic ow past a pointed cone 7 Comparison of theoretical and experimental results at Mach 8 7 DLR research report PB 9202 1992 21Stewartson K The theory of laminar boundary layers in compressible uids Oxford University Press 1964 22Shapiro A H The Dynamics and Thermodynamics of Compressible Fluid Flow Vol 1 The Ronald Press Company New York 1953 23Hein S Bertolotti F P Simen M Hani A and Henningson D Linear nonlocal instability analysis 7 the linear NOLOT code 7 Tech Rep DLR7IB 223794 A56 Institut fur Stromungsmechanik DLR Gottingen 1994 2 1Rosenboom I Hein S and Dallm Cone Flows AIAA Paper 993591 1999 25Stetson K and Kimmel R On Hypersonic Boundary Layer Stability AIAA Paper 920737 1992 26Stetson K Thompson E R and Donaldson C J Laminar Boundary Layer Stability Experiments AIAA Paper 831761 1983 27Becker E Gasdynamih Vol 6 in Leitfaden der angeWandten Mathematik und Mechanik B G Teubner Stuttgart 1966 L n ann U In uence of Nose Bluntness on BoundaryiLayer Instabilities in Hypersonic 14 of 14 American Institute of Aeronautics and Astronautics

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